XI.—On Unconditional Convergence in Topological Vector Spaces

1969 ◽  
Vol 68 (2) ◽  
pp. 145-157 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Sufficient Conditions ◽  
Direct Proof ◽  
Vector Spaces ◽  
Unconditional Convergence ◽  
Partial Sums ◽  
Locally Convex Spaces ◽  
Topological Groups ◽  
Topological Vector Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex

SynopsisFor a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

Projections and embeddings of locally convex operator spaces and their duals

1984 ◽  
Vol 96 (2) ◽  
pp. 321-323 ◽  
Author(s):  
Jan H. Fourie ◽  
William H. Ruckle
Keyword(s):  
Dual Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Complemented Subspace ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

scholarly journals A few remarks on bounded operators on topological vector spaces

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 763-772
Author(s):  
Omid Zabeti ◽  
Ljubisa Kocinac
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Operators ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Topological Vector Spaces ◽  
Different Types ◽  
Locally Convex

We give a few observations on different types of bounded operators on a topological vector space X and their relations with compact operators on X. In particular, we investigate when these bounded operators coincide with compact operators. We also consider similar types of bounded bilinear mappings between topological vector spaces. Some properties of tensor product operators between locally convex spaces are established. In the last part of the paper we deal with operators on topological Riesz spaces.

On the Nonstandard Duality Theory of Locally Convex Spaces

1980 ◽  
Vol 32 (2) ◽  
pp. 460-479 ◽  
Author(s):  
Arthur D. Grainger
Keyword(s):  
Vector Space ◽  
Duality Theory ◽  
Dual System ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Nonstandard Model ◽  
Locally Convex ◽  
External Property ◽  
Convex Spaces

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

On C-sequences of operators

2003 ◽  
Vol 40 (1-2) ◽  
pp. 145-150
Author(s):  
B. Wang
Keyword(s):  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Operator Series ◽  
Locally Convex ◽  
Convex Spaces

Invariant results are established for a considerably general multiplier convergence of operator series where the operators are defined on arbitrary topological vector spaces and valued in arbitrary locally convex spaces.

scholarly journals Characterizations of collectively precompact and semi-precompact opterators on topological vector spaces

1979 ◽  
Vol 28 (2) ◽  
pp. 179-188 ◽  
Author(s):  
M. V. Deshpande ◽  
S. M. Padhye
Keyword(s):  
Subject Classification ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Totally Bounded ◽  
Primary 47 ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

AbstractCharacterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.

scholarly journals Webbed spaces, double sequences, and the Macke convergence condition

1999 ◽  
Vol 22 (3) ◽  
pp. 521-524
Author(s):  
Armando García-Martínez
Keyword(s):  
Convergence Condition ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Double Sequences ◽  
Localization Theorem ◽  
Topological Vector Spaces ◽  
General Frame ◽  
Locally Convex ◽  
Webbed Space ◽  
Convex Spaces

In [3], Gilsdorf proved, for locally convex spaces, that every sequentially webbed space satisfies the Mackey convergence condition. In the more general frame of topological vector spaces, this theorem and its inverse are studied. The techniques used are double sequences and the localization theorem for webbed spaces.

scholarly journals On strong lifting compactness, with applications to topological vector spaces

1986 ◽  
Vol 41 (2) ◽  
pp. 211-223 ◽  
Author(s):  
A. G. A. G. Babiker ◽  
G. Heller ◽  
W. Strauss
Keyword(s):  
Structural Properties ◽  
Weak Topology ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Hausdorff Spaces ◽  
Topological Vector Spaces ◽  
Completely Regular ◽  
Locally Convex ◽  
Convex Spaces

AbstractThe notion of strong lifting compactness is introduced for completely regular Hausdorff spaces, and its structural properties, as well as its relationship to the strong lifting, to measure compactness, and to lifting compactness, are discussed. For metrizable locally convex spaces under their weak topology, strong lifting compactness is characterized by a list of conditions which are either measure theoretical or topological in their nature, and from which it can be seen that strong lifting compactness is the strong counterpart of measure compactness in that case.

scholarly journals Locally convex spaces with fundamental sequences

1973 ◽  
Vol 14 (1) ◽  
pp. 96-100 ◽  
Author(s):  
Marc De Wilde
Keyword(s):  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Fundamental Sequence ◽  
Locally Convex ◽  
Additional Assumptions ◽  
Convex Spaces

After Dieudonné [3], many authors have considered locally convex topological vector spaces that admit a fundamental sequence of compact, precompact,… subsets. Their work has been essentially to identify them as strong duals of Fréchet-Montel spaces, under suitable additional assumptions (barrelledness or evaluability). However, it seems that these spaces have never been characterized without additional assumptions. That is the aim of the present paper.

scholarly journals On Homomorphisms, Open Operators and their Adjoints

2001 ◽  
Vol 8 (4) ◽  
pp. 823-844
Author(s):  
D. Zarnadze
Keyword(s):  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

Abstract The well-known A. Grothendieck's theorem on a homomorphism between locally convex spaces is generalized to the case of topologies which are incompatible with dualities. On the basis of this theorem, necessary and sufficient conditions are obtained for a weak homomorphism (resp. its adjoint operator, resp. its double adjoint operator) to be again a homomorphism in various topologies of the initial (resp. dual, resp. bidual) spaces. Some new classes of pairs of locally convex spaces satisfying these conditions are established. The results obtained have enabled us to reveal new properties of frequently encountered homomorphisms and weakly open operators, as well as to strengthen and generalize some well-known results.

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